well you're onto something with the deep cycle thing, but car batteries aren't deep cycle so aren't really meant to be entirely drained in one sitting.. they're in the "starting" category of batteries. Though it would work, you'd damage the battery quickly if you completely drained it. A "golf cart" type battery would be more appropriate... it will last longer than a non-deep cycle battery and of the deep cycle category will last long than a marine battery, but a marine battery would work. you'd need to couple that with the other goods though of course... which is probably more work than just buying a UPS (Basic UPS systems are pretty inexpensive these days...). some charging and regulating mechanism (that shuts off to not overcharge) and a power inverter.
If you're looking at solar powered systems one big difference between the setup of a solar system and a UPS type system is the idea that in a computer backup system the switchover time needs to be nearly instantaneous. This might be a hurdle you can't achieve. Unless it's a laptop you will also have to recondition the power's sin wave output and/or phase and all kinds of other fun stuff. A laptop is far more forgiving.
I'm guessing it is a cheaper/easier solution if you bought a cheap UPS then somehow reconfigured that UPS' battery to use your own. (Though a not-cheap UPS made by APC or TrippLite that has changeable batteries is probably much, much easier to modify than something encased in plastic completely).